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Theorem climre 12073
Description: Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
Hypotheses
Ref Expression
climcn1lem.1  |-  Z  =  ( ZZ>= `  M )
climcn1lem.2  |-  ( ph  ->  F  ~~>  A )
climcn1lem.4  |-  ( ph  ->  G  e.  W )
climcn1lem.5  |-  ( ph  ->  M  e.  ZZ )
climcn1lem.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climre.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( Re `  ( F `  k ) ) )
Assertion
Ref Expression
climre  |-  ( ph  ->  G  ~~>  ( Re `  A ) )
Distinct variable groups:    A, k    k, F    k, G    ph, k    k, Z    k, M
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
Allowed substitution hint:    W( k)

Proof of Theorem climre
StepHypRef Expression
1 climcn1lem.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climcn1lem.2 . 2  |-  ( ph  ->  F  ~~>  A )
3 climcn1lem.4 . 2  |-  ( ph  ->  G  e.  W )
4 climcn1lem.5 . 2  |-  ( ph  ->  M  e.  ZZ )
5 climcn1lem.6 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
6 ref 11591 . . 3  |-  Re : CC
--> RR
7 ax-resscn 8789 . . 3  |-  RR  C_  CC
8 fss 5362 . . 3  |-  ( ( Re : CC --> RR  /\  RR  C_  CC )  ->  Re : CC --> CC )
96, 7, 8mp2an 655 . 2  |-  Re : CC
--> CC
10 recn2 12068 . 2  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  CC  ( ( abs `  ( z  -  A
) )  <  y  ->  ( abs `  (
( Re `  z
)  -  ( Re
`  A ) ) )  <  x ) )
11 climre.7 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( Re `  ( F `  k ) ) )
121, 2, 3, 4, 5, 9, 10, 11climcn1lem 12070 1  |-  ( ph  ->  G  ~~>  ( Re `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    C_ wss 3153   class class class wbr 4024   -->wf 5217   ` cfv 5221   CCcc 8730   RRcr 8731   ZZcz 10019   ZZ>=cuz 10225   Recre 11576    ~~> cli 11952
This theorem is referenced by:  mbflim  19017
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956
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